Find the best tutors and institutes for Class 12 Tuition
Q5(iii):
Evaluate the determinant: (iii) $\begin{vmatrix} 0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0 \end{vmatrix}$
Solution :
Step 1: Establishing the Determinant Structure
To evaluate the determinant of a $3 \times 3$ matrix $A$, we apply the Laplace expansion method. Let $A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$. The determinant $|A|$ is calculated by expanding along any row or column. We shall expand along the first row ($R_1$).
The formula for expansion along $R_1$ is: $|A| = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})$
Step 2: Identifying Matrix Elements
Given the determinant $\Delta = \begin{vmatrix} 0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0 \end{vmatrix}$, we identify the coefficients:
- $a_{11} = 0, a_{12} = 1, a_{13} = 2$
- $a_{21} = -1, a_{22} = 0, a_{23} = -3$
- $a_{31} = -2, a_{32} = 3, a_{33} = 0$
Step 3: Calculating Minors and Expansion
Substituting the values into the expansion formula [Per the Definition of Determinant of a Matrix]:
$\Delta = 0 \begin{vmatrix} 0 & -3 \\ 3 & 0 \end{vmatrix} - 1 \begin{vmatrix} -1 & -3 \\ -2 & 0 \end{vmatrix} + 2 \begin{vmatrix} -1 & 0 \\ -2 & 3 \end{vmatrix}$
Calculating the $2 \times 2$ determinants:
- First term: $0 \cdot [(0)(0) - (-3)(3)] = 0 \cdot [0 - (-9)] = 0$
- Second term: $-1 \cdot [(-1)(0) - (-3)(-2)] = -1 \cdot [0 - (6)] = -1 \cdot (-6) = 6$
- Third term: $2 \cdot [(-1)(3) - (0)(-2)] = 2 \cdot [-3 - 0] = 2 \cdot (-3) = -6$
Step 4: Algebraic Summation
$\Delta = 0 + 6 + (-6)$
$\Delta = 6 - 6$
$\Delta = 0$
Final Solution: The value of the determinant is $0$.
More Questions from Class 12 Mathematics Determinants EXERCISE 4.1
- Q1: Evaluate the determinant: $\begin{vmatrix} 2 & 4 \\ -5 & -1 \end{vmatrix}$
- Q2(i): Evaluate the determinant: (i) $\begin{vmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{vmatrix}$
- Q2(ii): Evaluate the determinant: (ii) $\begin{vmatrix} x^2 - x + 1 & x - 1 \\ x + 1 & x + 1 \end{vmatrix}$
- Q3: If A = $\begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}$, then show that $| 2A | = 4 | A |$
- Q4: If A = $\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4 \end{bmatrix}$, then show that $| 3 A | = 27 | A |$
- Q5(i): Evaluate the determinant: (i) $\begin{vmatrix} 3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0 \end{vmatrix}$
- Q5(ii): Evaluate the determinant: (ii) $\begin{vmatrix} 3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1 \end{vmatrix}$
- Q5(iv): Evaluate the determinant: (iv) $\begin{vmatrix} 2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0 \end{vmatrix}$
- Q6: If A = $\begin{bmatrix} 1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9 \end{bmatrix}$, find $| A |$
- Q7(i): Find values of $x$, if (i) $\begin{vmatrix} 2 & 4 \\ 5 & 1 \end{vmatrix} = \begin{vmatrix} 2x & 4 \\ 6 & x \end{vmatrix}$
- Q7(ii): Find values of $x$, if (ii) $\begin{vmatrix} 2 & 3 \\ 4 & 5 \end{vmatrix} = \begin{vmatrix} x & 3 \\ 2x & 5 \end{vmatrix}$
- Q8: If $\begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix} = \begin{vmatrix} 6 & 2 \\ 18 & 6 \end{vmatrix}$, then $x$ is equal to
CBSE Solutions for Class 12 Mathematics Determinants
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Determinants
Super Tutor
Kaushambi, Ghaziabad
15 yrs of
Exp
32 students
1376 hrs of classes completed
300
per hour
Online Classes
Tutor's home
Teaching Experience β Physics & Mathematics (Class XI & XII) Position: Senior Secondary School Teacher Subjects Taught: Physics and Mathematics Institution: B.C.M. Arya Model School , Green Fields Public School Responsibilities & Achievements: Taught advanced concepts in Physics and Mathematics to senior secondary students (Classes XI & XII), aligning lesson plans with the CBSE/State Board curriculum. Simplified complex topics through conceptual clarity, real-life applications, and problem-solving techniques to enhance student understanding and performance. Conducted regular assessments, mock tests, and one-on-one doubt-clearing sessions to prepare students for board exams and competitive entrance tests (e.g., JEE, NEET). Developed customized study materials, assignments, and revision modules to cater to varied learning styles and academic levels. Fostered a disciplined yet interactive learning environment that encouraged analytical thinking and academic curiosity. Many students mentored under my guidance went on to secure top scores in board exams and excel in competitive examinations.
Thank you Maβam! the way you simplify difficult questions and use relatable examples are highly appreciated, your teaching style made my understanding of concepts crystal clear.
Top Tutor
D.D.Colony Shivam Road, Hyderabad
3 yrs of
Exp
3 students
44 hrs of classes completed
300
per hour
Online Classes
Tutor's home
I am an Engineer and hold a Masters degree in Thermal and Fluids Engineering. Teaching has always been a passion for me which is why, I have taken classes for students in Mathematics, Physics, etc. I have also taken classes in Mechanical Engineering subjects even during my years of academic studies.
The concepts are taught in a very easy way and are very interesting. I enjoy the class a lot. Thank you so much for your support. Highly recommended to all. Has good grip in the concept.
Top Tutor
Sector-40, Chandigarh
10 yrs of
Exp
3 students
12 hrs of classes completed
500
per hour
Online Classes
Student's Home
Tutor's home
m baljinder kaur professor in punjab university chandigarh. i have 10 years of experience. i taught 4 years in aakash institute to JEE and NEET level students also.
Baljinder ma'am teach very well , i love the way ma'am teach . I recommend everyone who is preparing for board exam , i was fail in my prebord exam , but when I joined ma'am class , after 2 months i got 88 percent in board. She encouraged a lot and that's y I got good score specially in organic chemistry, i was totally nil but now m very comfortable in that and due to this now m planning for jee exam with baljinder ma'am only. She can do anything. Strongly recommend.
Top Tutor
Gulshan Bellina Rd Greater Noida, Noida
10 yrs of
Exp
18 students
255 hrs of classes completed
500
per hour
Student's Home
My name is have a B.Com degree from graduation and an M.Com degree for master's . Apart from this, I have cleared the CA Inter course . I have been teaching accountancy subjects for the last 10 years to Class 11 & Class 12 students. There are so many students who took classes from me and have scored well in their exams . The only reason for this is that the students' concepts are cleared very well.
"Mujhe apne Accountancy teacher se kaafi kuch seekhne ko mila. Unka teaching style bahut hi engaging aur clear hai, jo har complex topic ko bhi simple bana deta hai. Har lesson practical examples ke saath hota hai, jo humare liye easily samajhna possible banata hai. Unke dedication aur passion ki wajah se hum Accountancy ko na sirf samajh paate hain, balki enjoy bhi karte hain. Best teacher ever!" βΊππ»
Top Tutor
Indian Institute of Technology Roorkee, Roorkee
5 yrs of
Exp
1 student
13 hrs of classes completed
600
per hour
Online Classes
Student's Home
I am working as an engineer and hold a PhD degree from IIT Roorkee. Last year, I successfully taught 6 students from this plateform.
Explain the topic very well . Understand the needs of students. Gave sufficient time to student for proper learning and understanding the subject.
